Meshes are imported. Textures are just images that don't even need any special import process, just "save". Materials are not imported, but created for the specific game/rendering engine.
Your specific questions, then addressing your TLDR:
Do applied Blender shaders work in game engines or are they only for
rendering directly from Blender?
Depending on what you mean by "applied", materials/shaders are specific to the rendering engine. There's not really any "export material". There are commonalities in how things are going to be handled, but if you haven't built, say, transmission into your game shader, you're not going to get transmission.
If so, do lighting effects (like emissive materials and specularity)
work in an equivalent way in other game engines?
No, not necessarily. Again, these are bits of materials, and your game shader decides how to handle them. If you don't build emission into your game shader, you don't get emission. (But emission, as Eevee does it, is the simplest shader there is.) There are multiple ways to handle specularity, and you decide how to handle it when you make your game shader(s).
Can you stack multiple maps (i.e. diffuse + specular) and then import
the targeted asset in another program?
Same theme: you can import textures, whether those be diffuse color maps, specular color maps, roughness/gloss/specular power maps, environment maps, or even IOR maps (for Fresnel.) But "import" here makes it sound more complicated than it is: textures are just image files, and you create them then save them. You can create your own game shader that uses any number of kinds of textures, but if you don't have a game shader that uses those kinds of textures, then there's no reason to save those images; textures are merely parameters for your game-engine materials, and if your game engine material doesn't read those parameters....
When you unwrap a mesh, can you import it into photoshop? Is there a
"best practice" to doing so? Any tutorials you'd recommend?
While I don't have PS, you generally cannot import 3D models into 2D image editing apps. You can import an exported UV map-- which is just a picture with outlines of your UV map edges, just an image, not something weird and magical that beginners sometimes think it is-- but painting on a 2D image with that as a guide is not often the best way to do things. There are other apps, most popularly Substance Painter, which will import 3D meshes, with their UV maps, and allow you to paint in 3D. This is similar to Blender's texture painting, except SP is a lot better (more powerful, cleaner.) There are free SP-clones, but I can't speak to their quality.
What is the standard game asset workflow for Blender?
high-poly mesh in Blender >
bake to low-poly mesh >
UV unwrap >
digital paint and maps in Photoshop >
apply maps, shaders and lighting in game engine (Unity/Unreal/whatevs)
That is a perfectly reasonable workflow. We might add "rig" in there, which would come before or after UV map (either is reasonable, and there are reasons for both, and you can always iterate.) A reasonably professional workflow is going to involve Substance Painter more than Photoshop. By "apply maps" etc, what is really meant is to design a material, which, yes, is game engine specific. A professional team will have both a shader coder and artists that makes maps, and the shader coder will likely end up creating a standard material and defining what maps need to be created by the artists. Realistically, a game is going to have a limited number of material templates to choose from-- possibly, only one, roughly comparable to Blender's Principled BSDF.
But perhaps it would be better to say that "apply shaders" etc is the first step, or parallel with make a high poly. You can't really make your textures without knowing how they're going to be used by your game engine. In some cases, you can make reasonable guess, a diffuse is a diffuse is a diffuse, but do you need a map of tangent rotation? Do you need an AO map? Do you need both an AO map and a cavity map? Those aren't questions you can answer until you know what your shader(s) are doing.
I said that is a reasonable workflow because it is not the only potential workflow. Sculpt, retopo, high-to-low is not the only way to make meshes and textures; you'll see different workflows from Call of Duty, Overwatch, Genshin Impact. The reality is, you adopt a workflow that suits your desired output, and your desired output is different from game to game.